The two estimates can be proven in the same way. The proof follows by using equation (REF ) after commuting it with \(L^m\)  (something that results in the appearance of the angular derivatives in the right-hand side – for fixed \(\ell \)  see equation (REF )) and then making use of Propositions REF  and REF  and that \(T = L + \underline{L}\) . The proof follows the same lines as the proof of Proposition 4.7 of [1]}.
